Questions tagged [normed-spaces]

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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Comparison of $L^p$ and $L^q$ norms to establish the inclusion between corresponding spaces

We can deduce that; for any $x \in \ell^p,$ the space of $p$-summable real sequences ($p \geq 1$), $$\lVert x \lVert_q \leq \lVert x \lVert_p,~p \leq q < \infty,$$ by just letting $e=\frac{x}{\...
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The norm in the n-simplex

İ have this question and my attempt to solve it as the following : Consider an n-simplex $[p_0, p_1,...,p_n]$. If $a,b \in [p_0, > p_1,...,p_n]$, then show that $||a − b|| \leq \sup_i ||a − p_i||$....
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What's the relation between (strict) convexity of unit balls and shortest distance paths in $l_p$ metric?

I'm reading the book Geometry of Quantum States by Bengtsson and Zyczkowski. They have a brief discussion on $l_p$ norms. Depending on circumstances, different choices of p may be particularly ...
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Tricky $L^p$ Spaces Inclusions [duplicate]

In sampling theory, $L^p$ spaces are very important mathematical constructs. To this, we are currently studying some of their properties in class. In this regard, one exercise that I just cannot wrap ...
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Schauder basis and convex combinations

Let us suppose that $(X, \lVert \cdot \rVert)$ is a normed space over $\mathbb{R}$ which has a Schauder basis, that is, there is a sequence of vectors $(x_n)_n$ in $X$ such that for all $x \in X$ ...
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Proving two norms are not equivalent. Counterexamples. $V = C([0,1])$, $||f||_{\infty} = max |f(x)|$, $||f||_{*} = max |xf(x)|$.

$V = C([0,1])$, $||f||_{\infty} = max |f(x)|$, $||f||_{*} = max |xf(x)|$. The question asks: if the two norms are equivalent? And if not, why? How do I deduce whether the two norms are equivalent or ...
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Compute sup norm of sequence of functions.

Consider the operator $T: C^2[0,1] \subset C^1[0,1] \to C^1[0,1]$ defined by $Tf=f'+f''$. Compute $\| T e^{-nx } \|_{\infty }$ and $\| T x^n \|_{\infty }$. My attempt. First I tried to compute $\| e^{...
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Practical uses of $p$ norms for $p\notin \{1,2,\infty\}$?

We all love normed spaces, but it seems like the $1$-, $2$-, and $\infty$-norms get the lion's share of the love. That's not admittedly not without good reason, but what of the other unsung norms with ...
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The space of Lipschitz continuous functions is dense in that of uniformly continuous functions?

Let $(X,d)$ be a metric space. Then The space of bounded uniformly continuous functions is dense in that of bounded continuous functions w.r.t. the supremum norm. ref The space of bounded Lipschitz ...
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For $p\ge 1$, $x\ge 0$ and $y\ge 0$, $|x^{\frac{1}{p}}-y^{\frac{1}{p}}|^p \le |x-y|$?

I vaguely remember seeing an inequality of the form: $$|x^{\frac{1}{p}}-y^{\frac{1}{p}}|^p \le |x-y|$$ for $p\ge 1$, $x\ge 0$ and $y\ge 0$. Is this correct? If so, how is it proven? Can it be ...
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Derivative of l2-norm w.r.t matrix

I have a matrix A which is size of mm, a vector b which is size of m1. I'd like to derivative of the following function f(A) w.r.t A: $$f(A)=||A*b||_2$$ I need to find the first order Taylor expansion ...
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Exercise 6.A.17 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. I am worried if my solution is ok.

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 6.A.17 Prove or disprove: there is an inner product on $\mathbb{R}^2$ such that the associated norm is given by $$||(x,...
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Condition for finite $\infty$-norm of a transfer function

I am reading "Feedback control theory" by Doyle, Francis and Tannenbaum DFT. Lemma 1 on page 16 states: the $\infty$-norm of a transfer function $G$ is finite iff $G$ is proper and has no ...
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name this property: $‖Ax‖=c‖x‖$

I was recently working with some matrices that had this property: $$||Ax||_1=(n-1)||x||_1,\quad \forall x \in \mathbb{R} ^n$$ where $n$ is number of columns in $A$. It was a useful feature for ...
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Proving $\lVert T\rVert <\infty $ for $Tf(x):=\frac{1}{\sqrt2}f\big(\frac{x}{2}\big)$ (with $L^2$ norm on $f$)

Given $$T:PC[0,1]\to PC[0,1]$$ $$Tf(x):=\frac{1}{\sqrt2}f\big(\frac{x}{2}\big)$$ I'm trying to prove that: $\lVert T\rVert <\infty $ where: $$ \lVert T \rVert = \sup\{\lVert Tf(x) \rVert \mid \...
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Derivative of Frobenius norm with respect to scalar

Can anyone explain to me why $$\frac{d}{d\theta }\left\|\textbf{Aw}-\theta(1-c)\textbf{1} \right\|^{2}= 0$$ is equivalent to $$2(1-c)\textbf{1}^T(\theta(1-c)\textbf{1}-\textbf{Aw})=0$$? I'm not very ...
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Frobenius norm of sums of matrix products

If we have a sum of products of two conformable matrices: $\sum_{i=1}^NA_i'B_i$, I would like to understand if the following is generally true: $$ \left\|\sum_{i=1}^NA_i'B_i \right\|\leq \max_{1\leq i\...
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Property of inner product. [closed]

Why $\langle \nabla g(x_2),\frac{1}{L}\lVert \nabla g(x_2)\rVert\rangle \geq \frac{1}{L}\lVert \nabla g(x_2)\rVert ^2$? I found this inequality in IMC solution of exercise 6: https://www.imc-math.org....
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boundedness of infinite norm

I'm reading Theorical Numerical Analysis of Atkinson & Han and I'm trying one excercise but I'm stuck. On $C^1[a,b]$, define $$ (f,g)_*=f(a)g(a)+\int_a^b f'(x)g'(x)\,dx,\quad\forall f,g\in C^1[a,b]...
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How to prove that normed space is complete?

$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end ...
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minimal induced norm on sobolev space

I am new to functional analysis and induced norms. I have an exercise question in my university course, and I am trying a lot to get a solution for it. It is a challenging question for me. Could ...
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Functional analysis - best approximation norm [closed]

I have no idea how to prove or disprove the parts of the below question. I intuitively feel they are correct but cannot form a rigorous mathematical proof for it. Given is a normed space E. Also, ...
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ways to calculate norm of the Bounded Linear Functional.

$\Lambda$ be a linear functional on $C([0,1])$ defined by $$ \Lambda(f) = \int_0^1 xf(x)dx \;\;\; \text{ for } f \in C([0,1]). $$ and use $\|f\|_{sup} = \sup_{x \in [0,1]} |f(x)| $ for $f \in C([0,1])$...
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On the Interpretation of Volterra's theory of Integral Equations

Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1) $$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous ...
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Complexity of Lebesgue measurable spaces

Consider a discrete finite set $\Omega=X\times Y \in \mathbb{R}^{m\times n}$ for finite $m,n$. Let $(\Omega,\Sigma,\mu)$ be the measure space. ($\Sigma$ is the power set and $\mu$ is $\sigma$-finite ...
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Compute the norm of a linear bounded operator

Define $T:(\ell_\infty,||.||_{\infty}) \rightarrow (\mathcal{C}[-1/2,1/2]),||.||_{\infty})$ by $T(x(n))=f(t)=\sum_{n=1}^\infty x(n)s^k$ show that is a continous operator and compute the norm of $T$ ($|...
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Dense subspace and the annihilator of a set.

Let $X$ be a linear normed space and $A$ a arbitrary subset then show that the closure of $\left<A\right>$ is dense iff the annihilator of $A$ is $\{0\}$ (the zero functional) I try to use the ...
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2 votes
1 answer
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Derivate of the Limit of sequence function in $\mathcal{C}'([0,1]$

Let $\{x_n\}_n \subset \mathcal{C}^1([0,1])$ such that $||x_n-x||_{\infty} \rightarrow 0$ and $||x_{n}^{'}-y||_{\infty} \rightarrow 0$ show that $y \in \mathcal{C}([0,1])$ and $x'=y$ I proved that $y ...
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Density of $[H^2(\Omega) \cap H^1_0(\Omega)] \times H^1_0(\Omega)$ in $H^1_0(\Omega) \times L^2(\Omega)$

I am reading Evans' PDE proof of theorem 6, section 7.4.3, page 444 where the following is said to be "clear" $[H^2(\Omega) \cap H^1_0(\Omega)] \times H^1_0(\Omega)$ is dense in $H^1_0(\...
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Closed mid-point convex set and it's usefulness

$(X, \|•\|) $ be any normed space and $E\subset X$ . $(I)$ For every sequence $(x_n), (y_n) \in E$ with $x_n\to x $ and $y_n\to y $ in the space $(X,\|•\|) $ implies $\frac{x+y}{2}\in E\space \space $...
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Checking out 2 proofs refered to Lipschitz-continuity

I'd like to check 2 proofs I'm using. Both were made after some discussions I read here, but I have doubts about whether I'm making right interpretations and whether I'm formulating correctly both ...
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How to prove the space $L^2\big([a,b],w(t)\big)$ is complete and separable?

Suppose $w(t)$ is a positive and measurable function on $[a,b]$. If $x(t)$ is a measurable real function on $[a,b]$ satisfying $$ \left\| x \right\|^2 =\int\limits_{\left[ a,b \right]}{w\left( t \...
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Upper bound for spectral norm of a random matrix

If we know $X$ is a $n\times n$ matrix, and each element has mean 0 and variance $b_{ij}^2$. We can also know the covariance $Cov(X_{ij},X_{kl})$. Is there any method to get the upper bound of $\...
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Are left and right operator norms equal?

If $V$ is a finite dimensional normed vector space and I identify row and column vectors then I can define the action of a matrix both from the left and from the right. I can define the left and ...
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2 answers
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$\Vert x \Vert^{2} + \Vert y \Vert^{2} \le \Vert x + y \Vert^{2} + \Vert x - y \Vert^{2} \le 4 \left( \Vert x \Vert^{2} + \Vert y \Vert^{2} \right)$

I need to prove that the following $\Vert x \Vert^{2} + \Vert y \Vert^{2} \le \Vert x + y \Vert^{2} + \Vert x - y \Vert^{2} \le 4 \left( \Vert x \Vert^{2} + \Vert y \Vert^{2} \right)$ is true in ...
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Proof that every Cauchy sequence converges in a complete Stonean Algebra

I'm working through Jech, Thomas J., Abstract theory of Abelian operator algebras: an application of forcing, Trans. Am. Math. Soc. 289, 133-162 (1985). ZBL0597.03030. and I fail to write a proof of ...
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3 votes
1 answer
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Is there a special name for a norm that satisfies $||x+y|| \leq \max\{||x||, ||y||\}$?

If we have a vector space $V$ with a norm $||\cdot ||$, is there a special name for it if it satisfies $$||x+y|| \leq \max\{||x||, ||y||\} $$ for any $x,y \in V$? Im writing up something and I need a ...
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1 vote
0 answers
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Difference between strong and uniform operator topologies

I am currently a little confused regarding the difference between the strong and uniform operator topologies. I know that, if $T , T_n : V \rightarrow W$, then $T_n \rightarrow T$ in the uniform ...
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2 votes
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What will be the next step to ensure that the pointwise convergence implies convergence in the operator norm?

$(X, \|•\|) $and $(Y, \|•\|') $ be two normed space and $\begin{align} {\scr{B}}{(X, Y) }&=\{T\in {\scr{L}}{(X,Y)} : T \text { is bounded } \}\end{align}$ $\|T\|_{op}=\sup_{\|x\|\le1}\|Tx\|' $ I ...
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2 votes
2 answers
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Is the Alexandroff one-point extension metrizable? [closed]

Let $(E,\tau)$ be a topological space and $\Delta\not\in E$; $E^\ast:=E\cup\{\Delta\}$ and $$\tau^\ast:=\tau\cup\underbrace{\left\{E^\ast\setminus B:B\subseteq E\text{ is }\tau\text{-closed and }\tau\...
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Is the operator norm of the derivative the same as the Riemannian norm?

On a Riemannian manifold, $T_p M$ is the set of all derivative operators at that point. The norm is given by $$\left\|\frac{d}{dx}\right\|=\sqrt{g\left(\frac{d}{dx}, \frac{d}{dx}\right)}=\sqrt{g_{i,j}\...
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$\infty$-norm of $G(s) \leq 1$-norm of $g(t)$?

I have the following question: Let $G(s)$ be the Laplace transform of $g(t)$. Prove that $||G(s)||_\infty \le ||g(t)||_1$ I have been doing it for some time now, and I just can't crack it. Can you ...
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Prove that the mean of two vectors has a lower norm than the vectors

The problem is: Let $E$ be a normed space. For any $x,y\in E$, with $x\neq y$ and $\|x\|=\|y\|$, show that $$ \left\|\frac{x+y}{2}\right\|<\min\{\|x\|,\|y\|\}. $$ When E is an inner product space ...
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Equivalent norms on $\mathscr{C}([0,1],\mathbb{R})$?

Let $E=\mathscr{C}([0,1],\mathbb{R})$, $\varphi\in E$, and $$\forall f\in E,\; \|f\|_\varphi=\displaystyle\int_0^1\left|f(t)\varphi(t)\right|\text{d}t$$ It is not to hard to see that $\|.\|_\varphi$ ...
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If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?

If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$? If is not true in general, please give some counter-...
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Notation for space of eventually-zero sequences

An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...
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1 answer
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Let $M\subseteq X$ be a maximal subspace of the normed space $X$. Is there a functional $f\in X^*$ such that $\ker f = M$?

Let $X$ be a normed space and let $M\subseteq X$ be a maximal subspace. Do we need $M$ to be closed in order to claim that there is a functional $f\in X^*$ such that $\ker f=M$? I proceeded as follows:...
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Subgradient with the Frobenius norm

I'm working in the space of symmetric positive semi-definite matrices $S_n^+$ considered as a Hilbert space with respect to the inner product $\langle A,B \rangle = Tr(A^t B)$. I'm computing a ...
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1 vote
1 answer
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continuous functions in strong(norm) topolgy and weak topology

While reading Wasserstein GAN paper and in Appendix A, it says that The norm topology is very strong. Therefore, we can expect that not many functions $\theta \mapsto \mathbb{P}_\theta$ will be ...
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9 votes
2 answers
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Example of an algebra that is a Banach space but not a Banach algebra

I'm looking for an example of a space $\mathbb{A} $ such, $\mathbb{A} $ is an algebra; $\mathbb{A}$ is equipped with a norm that makes it a Banach space; $\mathbb{A}$ is not a Banach algebra, i.e., ...
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